This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle, where following Thom's terminology, excellent means that no two critical points li…
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48 results for “excellent Morse functions”
Game of plates and olives counts Morse functions on a sphere.
problem Counting Morse functions on a sphere.
method Analyzing the game's rules and mapping to Morse functions.
result Confirming the speculation that log M_n ~ n log n.
The paper bounds the distortion of a quotient map on Riemannian manifolds.
problem Bounding the distortion of a quotient map on Riemannian manifolds.
method Using the Reeb construction and Gromov-Hausdorff distance, the paper provides bounds on the distortion involving the first Betti number and a novel thickness invariant.
result Bounds on the distortion of the quotient map involving the first Betti number and a thickness invariant.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Defines Morse-Bott functions on manifolds with boundary and proves inequalities.
problem No specific problem stated; focuses on generalizing Morse theory.
method Defines Morse-Bott functions and proves inequalities for manifolds with boundary.
result Proves Morse-Bott inequalities for manifolds with boundary.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
New method extends discrete Morse theory to simplicial complexes.
problem Discrete Morse theory on simplicial complexes.
method Morse shellings and compatible discrete Morse functions.
result Triangulated surfaces and manifolds have Morse shellable triangulations.
Pairs of Morse functionsmath.CA
The paper classifies pairs of Morse functions under different groups.
problem Classifying pairs of Morse functions in general position.
method Analysis of pairs of Morse functions under various groups.
result Classification of generic pairs of Morse functions, including quotients.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using -Betti numbers and functions describing critical points.
result Morse inequalities given in terms of -Betti numbers and group functions.
Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on .
result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on .
Defines concordance of Morse functions on manifolds and presents a condition.
problem Deciding if two Morse functions on the same manifold are concordant.
method Introduces concordance as a stronger equivalence relation than cobordism, and presents a necessary and sufficient condition for concordance.
result A necessary and sufficient condition for two Morse functions to be concordant is presented.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
New invariants study Morse functions' equivalence classes in persistent homology.
problem Understanding equivalence classes of Morse functions on spheres for persistent homology.
method Graph-equivalent and height-equivalent Morse functions, with fundamental moves.
result Established new invariants to discern Morse functions more effectively.
Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
Oriented area function is a perfect Morse function for polygonal linkages.
problem Understanding the topology of polygonal linkages.
method Generalization of oriented area function as a Morse function.
result Cyclic equilateral polygons are independent generators of configuration space's homology.
A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions …
Morse-Bott homologymath.AT
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular -cube chains when the function is constant. We show that the ho…
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…
Relative cup-length defined for non-Morse functions on manifolds.
problem Defining a lower bound on critical points of non-Morse functions.
method Using local Morse cohomology and cohomology of isolating neighborhoods.
result A lower bound on critical points stronger than absolute cup-length.
Study of Morse functions with constraints and their bordism groups.
problem Interpolating between Morse and generic functions' bordism groups.
method Elimination of cusps, Stein factorization, two-index theorem, handle extension theorem.
result Constrained bordism groups are related to connective bordism.
New Morse functions on curve moduli space via geodesics.
problem Understanding the moduli space of curves via geometric and combinatorial methods.
method Introducing Morse functions based on geodesic lengths and analyzing their critical points and indices.
result Found new explicit Morse functions on , leading to a combinatorial cell decomposition.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.
Morse theory extended to non-degenerate functions.
problem Smooth functions on compact Riemannian manifolds without nondegeneracy.
method Extension of Morse theory without nondegeneracy assumptions.
result Morse theory applies to functions with finitely many connected critical points.
This paper equips Morse cochain complexes with -algebra structures.
problem Equipping Morse cochain complexes with -algebra structures.
method Analogous to K. Fukaya's definition, this paper provides a detailed treatment of Abouzaid's approach.
result Provides a coherent and detailed treatment of Abouzaid's approach to Morse cochain complexes.
Proves strong Morse inequalities for area functional in low dimensions.
problem Proving Morse inequalities for area functional in specific dimensions.
method Analyzes area functional in codimension one, proving inequalities under given dimension constraints.
result Strong Morse inequalities for area functional in specified dimensions.
Let be a Morse-Bott function on a finite dimensional closed smooth manifold . Choosing an appropriate Riemannian metric on and Morse-Smale functions on the critical submanifolds , one can construct a Morse chain complex whose boundary operator is…
Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
The paper proves a theorem about torsion and Morse functions for covering spaces.
problem Proving a theorem about torsion and Morse functions for covering spaces.
method Using a fiberwise Morse function and the Novikov-Shubin invariant.
result Proves the Cheeger-Müller theorem for -analytic torsion form.
The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given…
On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one halfmath.DG
Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
problem Characterizing analytic functions on Banach spaces with specific gradient properties.
method Proof of converse to the Morse-Bott property for analytic functions on Banach spaces using Lojasiewicz gradient inequality and Morse Lemma.
result Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)…
Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.
problem Characterizing subgroups of automorphism groups of Kronrod-Reeb graphs.
method Analysis of diffeomorphisms preserving Morse functions on 2-torus.
result Full description of special classes of automorphism groups.
Research connects prime numbers, graph theory, and cohomology.
problem Understanding the distribution of prime numbers using graph theory and cohomology.
method Discrete Morse-Smale complex and cohomology analysis.
result Explicit relationships between prime counting functions and cohomological properties of graphs.
Paper studies gradient fields from discrete Morse functions for watershed-cut computation.
problem Computing watershed-cuts from discrete Morse functions.
method Discrete Morse Theory and simplicial stacks.
result Minimum Spanning Forest of dual graph is induced by gradient vector field.
Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.
problem Computing multiparameter persistence with new tools and methods.
method Adapting Forman's theory to vectorial setting and using combinatorial topological dynamics.
result Established more general result for sublevel sets and found a way to induce Morse decomposition.
Computes cusp cobordism groups for Morse functions on manifolds.
problem Understanding the cusp cobordism groups of Morse functions.
method Employed Levine's cusp elimination technique and created pairs of cusps along fold lines.
result Both unoriented and oriented cusp cobordism groups are cyclic of order two in even dimensions and infinite order in odd dimensions.
Proof that critical knots of Morse-Bott functions are graph knots.
problem Characterizing critical knots in Morse-Bott functions.
method Inductive proof on the number of index-1 critical knots.
result Critical knots of Morse-Bott functions are graph knots.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.
problem Understanding Morse functions on manifolds with boundaries.
method Constructing a cellular structure and analyzing its algebraic properties.
result Estimation of the number of critical points of a Morse function with boundary conditions.
In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
Morse Cellsmath.GT
CW decomposition of manifolds with Morse functions.
problem CW decomposition of manifolds with Morse functions.
method Generic gradientlike vector field and Morse function.
result Stable manifolds provide a CW decomposition.
Proves an analytical analogue of Morse's lemma for gradient fields near critical points.
problem Understanding the behavior of gradient fields near critical points of Morse functions.
method Proves an analytical analogue of Morse's lemma showing unique linear vector fields.
result Shows that gradient fields near critical points have a natural standard form.